In order to prove that the order of both \(\chi _{1,1}\)  and \(\chi _{4,2}\) 
is at least \(-1\)  we may use the restriction of the Teichmüller
modular form \(\chi _{4,0,-1}\)  constructed in [1]}.
It is known that it has a pole
of order 1 along the hyperelliptic locus in \(\mathcal {M}_3\) .
The relation between \(\chi _{4,0,-1}\)  and the pair
\((\chi _{4,-2}, \chi _{1,1})\) 
is provided by Remark REF .
From this we can conclude the result since the order of \(\chi _{1,1}\) 
satisfies a congruence condition.
But we shall give a direct argument that gives more information.
We may view \(\chi _{1,1}\)  and \(\chi _{4,-2}\)  as
meromorphic Picard modular
forms on \(\Gamma _1[\sqrt{-3}]\) . We start with
\(\chi _{1,1}=\left( \begin{matrix} \chi _{1,1}^{(0)} \\ \chi _{1,1}^{(1)}\\ \end{matrix}\right) \qquad \text{\rm with}\quad \chi _{1,1}^{(m)}=\sum _{n\ge r} f_n^{(m)} u^n \, .\) 
We assume that \(\chi _{1,1}\)  has order \(r\)  along \(T_1\) . Using the
action of \((1,1,\rho )\)  and \(-1_3\) 
we see that a non-zero term \(f_n^{(m)}\)  has
\(n\equiv m \, (\bmod \, 6)\) . In particular, by a slight variant
of Proposition
REF  a non-zero term \(f_r^{(m)}\) 
is a meromorphic modular form of weight \(1+m+r\)  on \(\Gamma _1(3)\) ,
regular outside (the orbit of) \(\tau _0=(1-\rho ^2)/3\) 
and with order at least \(r\)  at \(\tau _0\) .
The space \(M_1^{(1)}(\Gamma _1(3))\)  is generated
by \(\vartheta =\sum _{\alpha \in O_F} q^{N(\alpha )}\)  and we know that
\(M_r^{(r)}(\Gamma _1(3))\)  is generated by \(\vartheta ^r\)  as
explained at the end of Section .
This implies that \(f^{(m)}_r\)  is divisible by \(\vartheta ^r\) ,
so \(f_r^{(m)}=\vartheta ^r \varphi \) 
with \(\varphi \in M_{m+1}(\Gamma _1(3))\)  for \(m=0\)  or 1.
But then \(\varphi \)  is a non-zero multiple of \(\vartheta ^{m+1}\) ,
implying that
\(f_r^{(m)}\)  is a non-zero multiple of \(\vartheta ^{r+m+1}\) .
The anti-invariance of \(\chi _{1,1}\)  implies that \(r\equiv 0 \, (\bmod \, 6)\) .
Thus the order of \(\chi _{1,1}\)  equals the order of \(\chi _{1,1}^{(0)}\) .
Since holomorphic Picard modular forms have weight
\((j\ge 0,k \ge 0)\)  the order \(s\)  of
\(\chi _{4,-2}\)  along \(T_1\)  is negative.
If we write \(\chi _{4,-2}^{(m)}= \sum _{n\ge s} g_n^{(m)} u^n\) 
for \(m=0,\ldots ,4\) , then for non-zero \(g_n^{(m)}\)  we have
\(n\equiv m+3 \, (\bmod \, 6)\) .
Moreover, \(g_s^{(m)} \in M_{-2+s+m}^{(s)}(\Gamma _1(3))\) .
For non-zero \(M_{k}^{(s)}(\Gamma _1(3))\) 
we need \(-2+s+m\ge s\) , hence using the congruence restriction we see
\(m=2\)  for non-zero \(g_s^{(m)}\)  and it is a non-zero multiple of
\(\vartheta ^s\) .
We then have that
\(g_{s+1}^{(m)}=0\)  unless \(m=3\) , and in fact it is quasi-modular
and one observes that it is not zero by
applying the Equation REF  to
\(\vartheta ^{s}\) .
Similarly \(g_{s+2}^{(m)}=0\)  unless \(m=4\) . Using again
Equation REF  we see
\({\rm ord}(\chi _{4,-2}^{(4)})= {\rm ord}(\chi _{4,-2}^{(2)})+2\) .
The discriminant of \(f_4\)  and of \(f_4f_1\)  are invariants that define
scalar-valued modular forms. These invariants are given by
\(\Delta _4=J_{2,0,0}^3-6 J_{3,0,0}^2\) 
and \(\Delta _4 \, J_{1,4,0}^2\)  up to non-zero multiplicative scalars.
The weight of \(\nu (\Delta _4)\)  is 0 and that of \(\nu (J_{1,4,0})\) 
is 6, and these are units outside \(T_1\) . Therefore \(\nu (\Delta _4)\) 
is constant and \(\nu (J_{1,4,0})\)  must be a multiple of \(\zeta \) .
Since
\(J_{1,4,0}= a_0b_1^4-a_1b_0b_1^3+a_2b_0^2b_1^2-a_3b_0^3b_1+a_4b_0^4\) 
we see by looking at the term \(a_4b_0^4\)  that
\({\rm ord}(g^{(4)})+ 4 \, {\rm ord}(f^{(0)})= s+2+4\, r \ge 0\, ,\) 
where we write \(g^{(4)}=\chi _{4,-2}^{(4)}\)  and \(f^{(0)}=\chi _{1,1}^{(0)}\) ,
hence \(r\ge 0\)  and \(\chi _{1,1}\)  is holomorphic.
Since \(\dim M_{1,1}(\Gamma [\sqrt{-3}],{\det })=1\) 
we can identify \(\chi _{1,1}\)  with a non-zero multiple
of a generator \(E_{1,1}\)  of \(M_{1,1}(\Gamma [\sqrt{-3}],{\det })\) 
constructed in [2]}.
We conclude from the development given there
that it has order 0 with \(f_0^{(0)}\) 
being a non-zero multiple of \(\vartheta \)  and that \(\chi _{1,1}^{(1)}\) 
has order 1.
Using also the regularity of the
other terms of \(J_{1,4,0}\)  we find that
\({\rm ord}(g^{(0)},\ldots ,g^{(4)})= (\ge 1,\ge 1, \ge -1, \ge 0,\ge 1)\, .\) 
But using the congruence condition we see
\({\rm ord}(g^{(0)},\ldots ,g^{(4)})= (\ge 3,\ge 4, = -1, \ge 0,\ge 1)\, ,\) 
which proves that the order of \(\chi _{4,-2}\)  along \(T_1\)  equals \(-1\) .
